The lower bound in Theorem 4.1.2

Proof of (4.2). We use the notation above and work in the product manifold ¯M with the harmonic functionu(x, t) =e^√λtϕλ(x). For the purpose of the proof of (4.2), we will assume thatM isn−1 dimensional. All this is strictly for notational convenience and ease of presentation, as we will now work with the tubular neighbourhood of u, which then becomes n dimensional. Considering the tubular neighbourhood ofuinstead ofϕλdoes not create any problems because the nodal set ofu is a product, i.e.

{u= 0}={ϕλ= 0} ×R. (4.21)

As the tubular neighbourhoods we are considering are of at most wavelength radius and at the this scale the Riemannian metric is almost the Euclidean one, we have

T_u,^δ

2 ⊆Tϕ,δ×R. (4.22)

Hence, to obtain a lower bound for|Tϕ,δ|it suffices to bound|T_u,^δ

2|below. To this end, we consider a strip S:=M×[0, R0] where R0>0 is sufficiently large.

We will obtain lower bound on|Bⁱ√^r

λ(pi)∩T_u,^δ

2|, which will give the analogous statements for (4.19) and (4.20) for the function u. As mentioned before, depending on the doubling exponent of uin the ball Bⁱ√^r

λ(pi) we distinguish two cases, and we will prove that ^{|Tu, δ2}_ρn−1^∩B(x,ρ)δ ^| ≥c in the case of controlled doubling exponent, and ^{|Tu, δ2}_ρn−1^∩B(x,ρ)|δ ≥ N^1ε, whereN .√

λin the case of high doubling exponent.

Case I : Controlled doubling exponent:

In the regime of controlled doubling exponent, in which case it is classically known that the nodal geometry is well-behaved, we essentially follow the proof in [JM09]. Let B :=B(p, ρ) be a ball such thatu(p) = 0 anduhas bounded doubling exponent onB(p, ρ), that is, ^sup_sup^B(p,2ρ)|u|

B(p,ρ)|u| ≤C (ultimately we will setρ∼^√1_λ). Then, by symmetry results (see Proposition 6.4.1 below), we have that C1< ^|B_|B⁺−^||< C2, where B⁺={u >0} ∩B, B⁻={u <0} ∩B.

Letδ:= ˜cρ, where ˜cis a small constant to be selected later. Denoting byB_δ⁺theδ-neighbourhood ofB⁺, and similarly forB⁻, and 2B:=B(p,2ρ), we have that sinceTu,δ⊃B_δ⁺∩B_δ⁻,

|Tu,δ∩2B| ≥ |B⁺_δ|+|B_δ⁻| − |B(p, ρ+δ)|. (4.23) By the Brunn-Minkowski inequality, we see that|B⁺_δ| ≥ |B⁺|+nωn^1/nδ|B⁺|^1−1/n, whereωn is the volume of then-dimensional unit ball. Setting|B⁺|=α|B|,|B⁻|= (1−α)|B|, we have

|Tu,δ∩2B| ≥ωn

ρⁿ−(ρ+δ)ⁿ+nρⁿ⁻¹δ(α^1−1/n+ (1−α)^1−1/n)

. (4.24)

By asymmetry,αis bounded away from 0 and 1, meaning thatα^1−1/n+ (1−α)^1−1/n>1 +C.

Now, taking ˜c small enough, the right hand side of (4.24) is actually&ρⁿ⁻¹δ, giving us

Lemma 4.3.1. Let the tubular distanceδand the radius of the ballρbe in proportion ^δ_ρ ≤c˜where

˜

c >0is a small fixed number. Assume that the doubling index ofuover the ballBρ is small. Then

|Tu,δ∩2B|&ρⁿ⁻¹δ. (4.25)

Case II: Large doubling exponent:

Now, let us consider a ballB(p, ρ) with radiusρ comparable to the wavelength, and letB⁰ = B(p,^ρ₂). Let us assume that initially we takeρsuch that ^δ_ρ ≤˜c.

Suppose _sup^supB0|u|

1

2B0|u| is large. By (4.14) and (4.15), the frequency function N(p,^ρ₂) is also large.

Recall also the almost monotonicity of the frequency functionN(x, r), given by (4.16), which will be implicit in our calculations below.

We will make use of the following fact:

Theorem 4.3.1. Consider a harmonic functionuonB(p,2ρ). IfN(p, ρ)is sufficiently large, then there is a number N with

N(p, ρ)/10< N <2N(p,3

2ρ). (4.26)

such that the following holds: Suppose that ∈ (0,1) is fixed. Then there exists a constant C = C() > 0 and at least [N]ⁿ⁻¹2^{ClogN/log logN} disjoint balls B(xi,_Nlog^ρ6N) ⊂ B(p,2ρ) such that u(xi) = 0. Here[.]denotes the integer part of a given number.

Theorem 4.3.1 is a straight-forward modification of the methods in Section 6 of [Log18b] - for completeness and convenience, we give full details of the proof of Theorem 4.3.1 in Section 4.4.

We will now use Theorem 4.3.1 in an iteration procedure. The first step of the iteration proceeds as follows.

Let us denote byζ1 the radius of the small balls prescribed by Theorem 4.3.1, i.e.

ζ1:= ρ

Nlog⁶N. (4.27)

Further, letB¹denote the collection of these small balls insideB(p,2ρ). LetF1:= inf_B∈B1

|Tu,δ∩B|

ζ₁ⁿ⁻¹δ

and let us assume that it is attained on the ballB¹∈ B¹.

We then have that

|Tu,δ∩B(p,2ρ)| ≥ X

B_i∈B1

|Tu,δ∩Bi| ≥[N]ⁿ⁻¹2^{ClogN/log logN}F1ζ₁ⁿ⁻¹δ≥

≥[N]ⁿ⁻¹2^{ClogN/log logN} ρⁿ⁻¹δ

(2Nlog⁶N)ⁿ⁻¹F1, which implies that

|Tu,δ∩B(p,2ρ)|

ρⁿ⁻¹δ ≥2^{ClogN/log logN}F1≥F1, (4.28) by reducing the constant C, if necessary, and assuming thatN is large enough. Recalling that by assumptionF1=^|Tu,δ_ζn−1^∩B1|

1 δ , we obtain

|Tu,δ∩B(p,2ρ)|

ρⁿ⁻¹δ ≥2^{ClogN/log logN}|Tu,δ∩B¹|

ζ₁ⁿ⁻¹δ . (4.29)

This concludes the first step of the iteration.

Now, the second step of the iteration process proceeds as follows. We inspect three sub-cases.

• First, suppose thatδandζ1are comparable in the sense that 8δ

ζ1

>˜c, (4.30)

where ˜c is the constant from Lemma 4.3.1. As there is a ball of radiusδ centered atx1 (the center ofB¹) that is contained in the tubular neighbourhood, we obtain

|Tu,δ∩B¹|

ζ₁ⁿ⁻¹δ ≥ C(˜cζ1)ⁿ

ζ₁ⁿ⁻¹δ ≥Cc˜ⁿζ1

δ. (4.31)

Furthermore, initially we assumed that ^δ_ρ ≤˜c, hence

|Tu,δ∩B¹|

ζ₁ⁿ⁻¹δ ≥C1˜cⁿ⁻¹ζ1

ρ =C1˜cⁿ⁻¹ 1

Nlog⁶N ≥C2

1

N¹, (4.32)

where1>0 is slightly larger than . In combination with the frequency bound of Lemma 4.2.1 and the fact thatN is comparable to the frequency by (4.26) we get

|Tu,δ∩B(p,2ρ)|

ρⁿ⁻¹δ ≥|Tu,δ∩B¹| ζ₁ⁿ⁻¹δ ≥ C3

λ^1/2. (4.33)

The iteration process finishes.

• Now suppose that the tubular radius is quite smaller in comparison to the radius of the ball,

i.e. 8δ

ζ1 ≤˜c. (4.34)

Suppose further that the doubling exponent ofuin ¹₈B¹is small. We can revert back to Case I and Lemma 4.3.1 by which we deduce that

|Tu,δ∩B(p,2ρ)|

ρⁿ⁻¹δ ≥|Tu,δ∩B¹|

ζ₁ⁿ⁻¹δ ≥|Tu,δ∩¹8B¹|

ζ₁ⁿ⁻¹δ ≥C, (4.35)

whence the iteration process stops.

• Finally, let us suppose that

8δ

ζ1 ≤˜c, (4.36)

and further that the doubling exponent ofuin B¹ is sufficiently large. We can now replace the initial starting ballB(p,2ρ) byB¹and then repeat the first step of the iteration process for ¹₈B¹. As above, we see that there has to be a ball ˜B¹of radius ˜ζ1∈(¹₄ζ1,¹₂ζ1) upon which the frequency is comparable to a sufficiently large numberN1. Now, we apply Theorem 4.3.1 and withinB¹ discover at least [N₁]ⁿ⁻¹2^{ClogN1/log logN1} balls of radius

ζ2:= ζ1

N₁log⁶N1

, (4.37)

such thatϕλ vanishes at the center of these balls.

As before, we denote the collection of these balls by B² and put F2 := infB∈B2

|T_u,δ∩B| ζ₁ⁿ⁻¹δ . Analogously we also obtain

F1= |Tu,δ∩B¹|

ζ₁ⁿ⁻¹δ ≥2^{ClogN1/log logN1}F2≥F2. (4.38) Again, we reach the three sub-cases. If either of the two first sub-cases holds, then we bound F2 in the same way asF1 - this yields a bound on ^|Tu,δ_ρn−1^∩Bδ^ρ|. If the third sub-case holds, then we repeat the construction and eventually getF3, F4, . . ..

Notice that the iteration procedure eventually stops. Indeed, it can only proceed if the third sub-case is constantly iterated. However, at each iteration the radius of the considered balls drops sufficiently fast and this ensures that either of the first two sub-cases is eventually reached.

This finally gives us

|Tu,δ∩B(p,2ρ)|

ρⁿ⁻¹δ ≥F1≥F2≥ · · · ≥ C3

λ^1/2. (4.39)

At last, we are done with the iteration, and this also brings us to the end of the discussion about Case I and Case II. To summarize what we have established, the most “unfavourable” situation is that scenario in Case II, where we at every level of the iteration we encounter balls of high doubling exponent, and we have to carry out the iteration all the way till the radius of the smaller balls (whose existence at every stage is guaranteed by Theorem 4.3.1) drops below δ. The lower bound for ^|Tu,δ_ρ^∩B(p,2ρ)|n−1δ in such a “worst” scenario is given by (4.39).

We are now ready to finish the proof. Lettingρ= ₂^√r_λ and by summing (4.39) over the∼λ^n/2 many wavelength ballsBⁱ√^r

λ

(as mentioned at the beginning of this Section), we have that

|Tu,δ| ≥ C3

λ^1/2ρⁿ⁻¹δλ^n/2&λ^1/2−1/2δ. (4.40)

Using the relationship between the nodal sets ofϕλ andu, this yields (4.2).

Im Dokument On the geometry of nodal sets and nodal domains (Seite 78-82)